computer-aided design diagnosis for machines—kinematic model extraction from mechanisms

Mech. Mach. Theory Vol. 25, No. 3, pp. 355-364, 1990 0094-114X/90 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1990 Pergamon Press plc

COMPUTER-AIDED DESIGN DIAGNOSIS FOR MACHINES--KINEMATIC MODEL EXTRACTION

FROM MECHANISMS

ATSUSHI NAKAMURA I and NAOMASA NAKAJIMA 2 ~Body Design Department, Nissan Motor Company and 2Department of Mechanical Engineering,

Faculty of Engineering, University of Tokyo, Tokyo, Japan

Abstract--This paper presents a method for kinematic model extraction from a given mechanism as an approach to design diagnosis. A computerized method of extracting a kinematic model, which is usually expressed by a kinematic diagram, is proposed and illustrated with four mechanisms of 4--34 elements. Also verification of intended kinematic models, and prediction of element separation in faulty mechanisms were successfully carried out. As a unified expression method for the mechanism and kinematic model, the Composition Diagram is proposed in our study.

1. INTRODUCTION

This paper presents a method of computer-aided design diagnosis for mechanisms and machines. The word "design diagnosis" in the paper means detection of defects in the object being designed. The purpose of design diagnosis is to help the designer in realizing a sound and reliable design.

We think design diagnosis aided by the computer is becoming indispensable in present machine design. Machine designers have striven to design perfect machines and made efforts to eliminate defects by adopting various means, like design check manuals. However, recently the complete detection of design defects has become more and more difficult as machines become more complex. Nevertheless the quest for high reliability is increasing.

It is true that some defects might be detected during manufacturing or in the working test of the machine. However, defects in design should be detected during the design process, because the subsequently required redesign will consume much time and resources. Also, defects concealed in the product may cause an accident or a disaster.

Diagnosis in machine design has intrinsically difficult problems compared with electronic circuit design. In LSI circuit design the design specification is well-described logically and the circuit is designed in a way that its function, i.e. the logical performance coincides with the specifications. Then diagnosis can be treated by comparison between the design specification and the result of a computer simulation of the designed circuit.

On the contrary the specification of a machine design is difficult to describe by logical expressions alone, because the function of a machine mainly depends upon actual shapes of the machine components, that is, it is shape-dependent. The function is mainly carried by a three-dimensional shape and layout of each element. A design diagnosis process in machine design must be able to handle shapes.

In this paper we will explain a computerized method of extracting a kinematic model from a given mechanism. The term "kinematic model" implies kinematic functions usually represented by a kinematic diagram or skeleton.

In our study shapes and composition of a machine for diagnosis are described using the Feature Description (FD) language and then input to a computer, The Feature Description language was developed at the laboratory of one of the authors [1]. The extracted kinematic model as a result is presented using a graphical expression named Composition Diagram as proposed in our study.

One of the pioneering attempts of computerized shape-dependent diagnosis of a designed machine was the work done by Ishida [2]. His work focussed on detecting unanticipated functions in a design, showed the feasibility of detecting, for example, hydraulic leakage in a cylinder, locking of a shaft and a bearing caused by thermal expansion, and undissolvability of an assembly.

355

356 ATSUSHI NAKAMURA and NAOMASA NAKAJIMA

A systcmatic approach for shapc-dcpcndcnt diagnosis was followed. The proccdurc of automatic gcncration of kinetic and kinematic equations for a gcncral plane link mechanism was proposed by Nakajima et al. [3]. Also, a mcthod of expressing functions of a machinc and a gcncral proccdurc of functional simulation wcrc shown precisely by Murakami and Nakajima [4].

Our study is intended to compensate for limitations of thc last two works. In the generation of thc equations thc information of the kinematic model extracted from the mechanism in question must bc input. The functional simulation cannot answer cxplicitly whcthcr or not the rcquircd kinematic model is involved in the given mechanism, because the simulation treats a mechanism as a black box and only output the performance of it.

2. EXTRACTING KINEMATIC MODEL

2.1. Basic concept of extracting kinematic model

The procedure of extracting a kinematic model from a given mechanism in this study is basically similar to the designer's typical procedure. He first reads a drawing of a mechanism and identifies each element. Second he analyzes the relationship between elements with respect to their mutual movement, then joins the mutually unmovable elements together considering them as one element.

(a) 3 2 1 (13

Y

(b) P PCC 1 P C P 2 ~ [ ~

In~,, , ~ j Iq'--l-p ~11 [] II 1 I , l I

) (c) P PCC I P ~CP 2 P PCC 3

(d) PPPPCC I PPP~CP 2 ~PPp(~(: : ~

"11 ~

(e) RP RP 1 RP RP 2 RP RP 3

Fig. 1. Explanation of the Composition Diagram.

Computer-aided design diagnosis for machines 357

ta)

p p 1 p ~ 2

1 p 1 ~ 2

J

I (b)

I DII Dol

I t 1 ¢ , 2

Fig. 2. An example of applying Procedure 1.

After analyzing the composition of these elements, he suggests a kinematic model corresponding to the given mechanism.

In our computerized procedure the drawing of a mechanism is initially translated into a verbal description by the designer using the Feature Description language. This description holds the information of the mechanism with respect to shapes, dimensions and mating surfaces of whole elements. In other words the description holds a list of contact dements including the information of mating surfaces.

Then the computerized process consists of the following three steps:

(1) deducing the contact list from the mechanism description; (2) analyzing relationships of indirect contacts between elements; (3) arranging the set of mutually unmovable elements into a single element; (4) extracting pairs of elements and deducing a kinematic model.

In order to treat the above procedure we introduce a diagram type called the Composition Diagram which is explained in the next section.

2.2. Some restrictions on mechanisms under consideration

Before explaining the extraction procedure in detail, we must define the mechanisms to be treated in this study. Conditions and restrictions on the mechanisms are listed below.

(1) The mechanism is composed of several elements which are considered to be rigid bodies, not divisible into smaller parts.

(2) The type of contact between elements is restricted to one of the following four types:

plane to plane; cylinder (rod) to cylinder (hole); screw (male) to screw (female); gear (outer) to gear (outer or inner).

3.58 ATSUSHI NAKAMURA and NAOMASA NAKAJIMA

(3) The contact plane is normal to one of the axes of the Cartesian coordinate, which is defined for each element, and the center axis of the cylinder, the screw, or the gear is parallel to one of the co-ordinate axes.

The kinematic model of the mechanism is expressed by one of the following six pairs or combination of them:

rotating pair; prismatic pair; cylindrical pair; plane pair; screw pair; gear pair.

3. THE COMPOSITION DIAGRAM

The Composition Diagram provides a unified expression of a mechanism and a kinematic model by means of a graphical representation. It holds information of the type and the position of contact of elements. Figure l(b)-(e) show an example of the Composition Diagram which represents a simple mechanism, two plates with a loose rivet, shown in Fig. l(a). The three blocks with a serial number represent elements, and the serial number corresponds to an element number. Three small rectangles inside each element block, the nodes of the diagram, represent the contact surface and the direction of the normal vector of the surface. The positions of the nodes, the left, the middle and the right correspond to the direction of the x, y and z axis, respectively. The lines, the arcs of the diagram, represent the contact relationships. The type of contact surface is expressed by the symbol above the element block. The symbols are as follows:

P,/~--positive, negative plane with respect to the direction of the normal vector; C,(~--~ylindrical column, hole; S,S--male, female screw; G,tT--outer, inner gear.

The diagram of Fig. l(b) shows relationships of direct contacts deduced by interpreting the mechanism description using the Feature Description language. The diagram is successively changed and finally transformed into the kinematic model in Fig. l(e). The process will be explained later.

In the expression of the kinematic model, the meaning of the blocks and nodes in the diagram changes. The block with a serial number represents an element which is correspoding to a set of

Computer

Frame structured database

I Procedures for diagram transformation

I Procedures for diagram representation u ~ Display Composition Diagram

Fig. 3. Basic structure of the kinematic model extraction system.


mutually unmovable elements. Each node represents one element of pairs in kinematics. That is, one couple of nodes linked by an arc represents one pair. The symbols of pairs are as follows:

RP--rotating pair; P R P - - p r i s m a t i c pair;

CP--cylindrical pair; PP--plane pair; SP--screw pair; GP--gear pair.

The expression of the kinematic model by a Composition Diagram in Fig. l(e) is equivalent to the common kinematic diagram in Fig. l(f).

Now, the problem of extracting the kinematic model from the given mechanism reduces to how the Composition Diagram of the mechanism [Fig. l(b)] can be transformed to the kinematic model [Fig. l(e)].

4. TRANSFORMATION PROCEDURES FOR THE COMPOSITION DIAGRAM

Transformation from the mechanism to the kinematic model is processed automatically by applying the following four procedures.

Procedure 1. If there is a lone surface with no contact to others or a contact surface parallel to another in the same element, then eliminate it.

Procedure 2. If there is virtual contact between any couple of elements, then add nodes and a link to the two elements.

Procedure 3. If there is a couple of elements which have no mutual movements, then put the couple together and eliminate any one of them.

Procedure 4. If there is no possibility of applying Procedure 1-3, then replace the relationship between two elements with the corresponding kinematic pair.

Each procedure is usually applied in numerical order repeatedly. Figure 2 shows the results of Procedure 1 for two examples, where multiple planes or cylinders in stepped relationship are changed to a single plane or cylinder.

Procedure 2 is the most important one. Virtual contact means the constraint of freedom caused by other elements which have no direct contact to the one in question. Using Fig. 1 again we will show the transformation of the diagram by applying Procedure 2. In Fig. l(c) a virtual link of thick line is added between element 1 and 2 with two virtual nodes C and t~. It was caused by the co-axial relationship between element 1 and 2 which was not expressed explicitly in the initial Composition Diagram in Fig. l(b). Three other virtual links with nodes are added in Fig. l(d). They were caused by a loop of constraints with respect to freedom of movement in the x direction.

The initial diagram is described based only on direct contact. The complete and explicit expression of the relationship between the two elements is given by adding a virtual link.

The introduction of the virtual link makes it easy to judge if any couple of elements is considered to be joined or not. Table 1 lists the possible combinations of links between two elements which are treated as joined. In the table a link is expressed by a pair of contact surfaces like P P or Ct~. The second row of the table represents:

if there are three links PP, P P and C(~ in the direction of one of the x, y, z directions and

there is a link P P or two links P P and P P in one of the other directions between the two elements in question,

Table 1. Possible combinations of links between two elements treated as joined Case A axis (one of x, y, z) B axis (excluding A) C axis (rest)

1 PP + PP PP + PP pl ~ + Pp 2 PP + P1J + C~ PP, PP + PP * 3 PP + PP + C~ + C~ * * 4 F * *


Table 2. Correspondences between a pair in a kinematic model and links between elements of mechanism

Pair A axis (one of x, y, z) B axis (excluding +4) C axis (rest) Rotating p PP + PP + CC None None Prismatic p pp + p]~ p]5 + pp None

C• PP PP, Pt 5 + PP, none CC PP + PP PP + PP, none C~ + CC PP PP, PP + PP, none C~ + CC PP + PP PP + PP, none

Prismatic pf PP + PP PP + PP PP Pa n + CC PP PP, PP + Pt ~, none PP + C~ PP + PP PP + PP, none PP + CC + C~ None None PP + CC + CC PP PP, PP + PP, none PP + C~ + CC PP + PP PP + PP, none

Cylindrical p CC None None Cylindrical Pt PP + CC None None Plane p PP + PP None None Plane Pt PP PP None

PP + PP PP None Screw p S~q None None Gear p GG, G~7 None None tMovement is restricted to single direction or half plane.

then the two elements are treated as joined.

The symbol "*" means any links including none, and " F " means the fastened relation, for example, a bolt and nut, or a fixed fit pin and hole.

Correspondences between a pair in a kinematic model and links between elements are listed in Table 2. The table used in Procedure 4.

5. I M P L E M E N T A T I O N

The basic structure of our system is shown in Fig. 3. Input is a description of the mechanism using the Feature Description language. The description is interpreted (by the interpreter) and stored as Composition Diagram data in the frame structured database, which is based on frame theory [5] and structured by using Frame Representation language [5]. The procedures, transform- ing the mechanism to the kinematic model and presenting diagrams, are written in the LISP language and stored in the same database. The contents of the Composition Diagram data changes as the transformation procedure progresses, and finally the data represent the kinematic model. We adopted the frame structured database because successive updating and handling of data is easy. The interpreter is also written in LISP.

The system runs on a personal computer, PC9801(NEC), and the programming language is mu-LISP86 on MS-DOS.

6. EXAMPLE OF KINEMATIC MODEL EXTRACTION

We will show in detail the results of a kinematic model extraction for a slider-crank mechanism as an example. Figure 4 shows the drawing of the slider-crank mechanism composed of nine elements. Figure 5(a) is the initial Composition Diagram of the mechanism. The diagram is the direct interpretation of the description of the mechanism. The broken lines mean links of fastened relation. Procedures 1 and 2 are applied to the initial Composition Diagram, and as a result all the virtual links and nodes are added [Fig. 5(b)]. Figure 5(c) is the diagram after Procedures 1-3 have been applied repeatedly. The number of elements is decreased to four. The final Composition Diagram which represents the kinematic model is obtained by applying Procedure 4 [Fig. 5(b)]. The diagram is equivalent to the common kinematic diagram [Fig. 4(e)]. The Composition Diagrams of Figs 5(a)-(d) were produced by the computer.


Fig. 4. Slider-crank mechanism for kinematic model extraction.

These diagrams show that the extraction of the kinematic model was processed correctly and successfully.

We also examined the following three mechanisms:

joint of a link (4 elements); vice (7 elements); positioning table (34 elements);

and obtained a correct kinematic model for each mechanism. Computation time for the extractions are listed in Table 3. Computation time seems to increase

in proportion to the 2rid power of the number of elements. We believe however, that computation time will not increase impractically, because the composition of a practical mechanism is generally hierarchical and at most a few hundred of elements will be processed simultaneously. Moreover computation time in Table 3 is easily reduced to one-hundredth if the personal computer is replaced by a suitable engineering workstation.

7. D E S I G N D I A G N O S I S

We will show the applications of the kinematic model extraction for design diagnosis in two different ways. The first way is a verification of a mechanism with respect to its kinematic model. This is easily realized by introducing a function of comparison between the intended kinematic model and the extracted one.

Table 3. Computation time of kinematic model extraction by PC9801

Mechanism Number of elements Computation time (m)

Joint of link 4 1 Viced 7 4 Slider-crank 9 5 Positioning table 34 90

362 ATSUSH] NAKAMURA and NAOMASA NAKAJIMA

i ' , ' ; : - - ~ - - - ; , - ' - L . i ~ - ~ " . - . ~ ; ; . - . ~ i " . ~ - ~ : i - : - - . = - . i . : i . . . . . : - : , :~:~ . . . . . . . . . . .~ - |

F I I I~1 ~ IIit1 ' " ( a ) : : -"-: i i t I I ~ - , L - - -~ I I , , ~ l l l l l I I I I ' ' "

~i ~ i ~,, , ,, J i l l [ I I

-Jl

(b)

o o I

(el

(e) !-

d

O O H 0 0 : 1 0 0 2 O 0 I ( ~ RPPP PRP RPPP _PPRP PPRP PRP

, i ] I 4

Fig. 5. Kinematic model extraction from a slider-crank.


Table 4. Cases when a single element may separate from the rest

Case A axis (one of x, y, z) B axis (excluding A) C axis (rest)

a Plane pair Plane pair, none None b Prismatic pair, cylindrical pair Plane pair, none Plane pair, none c Screw pair None None

Table 5. Combination of pairs in the case of a set of n elements when they may simultaneously separate from the rest

Element number A axis (one of x, y, z) B axis (excluding A) C axis (rest)

l Rotating pair, Prismatic pair, None plane pair cylindrical pair

i Rotating pair, None None (1 < i ~< n) plane pair

The second way is a prediction of successive separation of elements in the case of a missing or damaged element. The possibility of elements separating is easily judged by investigating the Composit ion Diagram of a kinematic model. There are two cases of separation:

(1) A single element may separate from the rest. (2) A set of more than one element may simultaneously separate from the rest,

however a single element cannot separate.

The conditions of the two cases are listed in Table 4 and 5. Geometrical interference is not considered in the conditions, so a ring on a finger, as an example, is treated as removable even in the direction of a finger base.

We have examined an example of a faulty system where a crank pin of a slider-crank mechanism (Fig. 4) is sheared and kept inside a connecting rod. We obtained a tree of possible successive separations as shown in Fig. 6 (element 5 is neglected). A branch in the tree indicates separation of elements and the numerals represent the corresponding element numbers.

In the example, the kinematic model of the faulty system was extracted first. The model is not the same as the one from the complete system. Note that the kinematic model holds the information of which elements in the mechanism are joined and treated as a single element in the kinematic model. Then the separating process was dealt with by the lists in Tables 4 and 5.

8. C O N C L U S I O N

An approach to design diagnosis by kinematic model extraction from mechanisms has been presented. With this work we have realized an effective method for design diagnosis. The work may be concluded as follows:

®'

Fig. 6. Tree of the possible successive separation in a faulty slider-crank.


(1) A compute r i zed m e t h o d o f ex t rac t ing a k inemat ic mode l f rom a given mechan i sm was sys temat ica l ly s tudied and the p rocedure was implemented on a persona l c o m p u t e r PC9801.

(2) The ex t rac t ion m e t h o d was eva lua ted by four mechan i sms o f 4--34 elements where co r r e spond ing k inemat ic mode l s were successfully extracted.

(3) Two types o f design d iagnos is a d o p t i n g the ex t rac t ion m e t h o d were tried: verif icat ion o f in tended k inemat ic mode l s and pred ic t ion o f e lement separa t ion in a faul ty system. Correc t results were ob ta ined in bo th types o f design diagnosis .

(4) The C o m p o s i t i o n D i a g r a m was devised and in t roduced in o rder to unify the express ion o f the mechan i sm and k inemat ic model .

R E F E R E N C E S

1. H. Takase and N. Nakajima, Proc. 1st Int. Symp. Des. Synthesis, Tokyo, pp. 600-605 (1984). 2. T. Ishida, H. Minowa and N. Nakajima, Proc. 1st Int. Symp. Des. Synthesis, Tokyo, pp. 21-26 (1984). 3. N. Nakajima, T. Murakami and K. Oikawa, Proc. Int. Conf. Engng Des., Boston, pp. 648-655 (1987). 4. T. Murakami and N. Nakajima, Artificial Intelligence in Engineering: Diagnosis and Learning (Ed. J. S. Gero),

pp. 199-226. Computational Mechanics Publications (1988). 5. P. H. Winston and B. K. P. Horn, LISP. Addison*Wesley, Reading, Mass. (1981).

C A D D I A G N O S E Fl~R M A S C H I N E N - - K I N E M A T I K - M O D E L L B I L D U N G Ft~R M E C H A N I S M E N

ZusammeafassMg--In diesem Artikel wird vine Vorgehensweise vorgeschlagen, nach dem ein kinematisches ModeU durch Rechneranwendung yon einem mechanischem Getriebe abgeleitet werden kann. Der Ausdruck "kinematisches Modell" beschreibt diejenige Funktionen, die iiblicherweise in einem kinematischen Diagram repr~.sentiert werden.

Das Ziel dieser Arbeit ist es, vine praktische Methode fiir die rechneruntersffitzte Konstruktions diagnose yon Maschinen-systemen anzubieten.

Die computerisierte Vorgehensweise zur Ableitung des kinematischen Modells besteht aus den folgenden vier Schritten:

1. Interpretation der verbalen Beschreibung vines Gertriebes und Auflisten der Elemente und deren gegenseitigen Beziehungen.

2. Zuftigen yon virtuellen Ikziehungen bedingt dutch indirekte Bindungen zwischen Elementen. 3. Zusammenstellen jeder Menge yon gegenseitig unbeweglichen Elementen zu einem Element. 4. Registrieren der Elementenpaare und Ableiten des entsprechenden kinetischen Modelles. Ftir die einheitliche Beschreibung dutch graphische Pr~.sentation vom Getrieben und kinematischen

Modellen wurde ein neuer Diagrammtyp, genant Composition Diagram, entwickelt. Um das Hantieren und Aufdatieren zu vereinfachen, wurde vine frame-strukturierte Datenbasis angewendet. Das System ist auf einem Personal-Computer PC9801 (NEC) aufgebaut, und die Programmiersprache ist mu-LISP86 auf MS-DOS.

Vier Getriebe mit 4, 7, 9 und 34 Komponenten wurden untersucht und die kinematischen Modelle vom Rechner korrekt abgeleitet. Die Resultate best~itigten, dass die Rechenzeiten ffir praktische Anwendungen nicht prohibitiv sind.

Auch wurde die entwickelte Vorgehensweise zur Voraussage des Verhaltens yon Getrieben mit einzelnen fehlenden oder defekten Komponenten erfolgreich angewendet.

computer-aided design diagnosis for machines—kinematic model extraction from mechanisms

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